bibkey: schulte2018a299406 authors: Werner Schulte year: 2018 title: “OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture” doi: null url: https://oeis.org/A299406 claim: “%N Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s)zeta(6s))/(zeta(2*s)zeta(3s)). %F Conjecture: a(n) = A008836(n) * A210826(n).” strata_touched:
- D5/S3/Arith/SchulteLiouvilleCubeMobius license: citation-only triage: anchor
OEIS A299406
Schulte’s coefficient identity relates A299406 to Liouville’s function and the Lambert-series coefficients A210826. Only the quoted product conjecture is the claim under consideration, for positive natural indices.
The coefficient reading of the Dirichlet generating function replaces
zeta(ks) by the k-th-power lift of the constant-one arithmetic function and
1/zeta(ks) by the k-th-power lift of the Möbius function. Thus A299406 is
zeta ⋆ liftPow 6 zeta ⋆ liftPow 2 mu ⋆ liftPow 3 mu.
The Lambert series for A210826 says that the sum of its coefficients over
the positive divisors of n is the indicator of a positive cube. Möbius
inversion gives mu ⋆ liftPow 3 zeta. These are coefficient interpretations;
analytic convergence of either series is outside the claim.
Verified locator
Readings dated 2026-09-15 (ASSUMED-UNVERIFIED):
- A299406 URL: https://oeis.org/A299406
- A299406
%N(verbatim): Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s)zeta(6s))/(zeta(2*s)zeta(3s)). - A299406
%O(verbatim): 1,1 - A299406
%F(verbatim): Conjecture: a(n) = A008836(n) * A210826(n). - A299406
%A(verbatim): Werner Schulte, Feb 20 2018 - A008836 URL: https://oeis.org/A008836
- A008836
%N(verbatim): Liouville’s function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity). - A210826 URL: https://oeis.org/A210826
- A210826
%N(verbatim): G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^3).
No literature proof was found in the searched surfaces recorded in the 2026-09-15 readings; these are scoped search results, not an exhaustive priority claim.